Question: All of the 5th grade teachers and students from Almond went on a field trip to an archaeology museum. Tickets were $$8.50$ each for teachers and $$4.50$ each for students, and the group paid $$52.50$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$34.00$ each for teachers and $$9.00$ each for students, and the group paid $$156.00$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8.5x+4.5y = 52.5}$ ${34x+9y = 156}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-34x-18y = -210}$ ${34x+9y = 156}$ Add the top and bottom equations together. $ -9y = -54 $ $ y = \dfrac{-54}{-9}$ ${y = 6}$ Now that you know ${y = 6}$ , plug it back into $ {8.5x+4.5y = 52.5}$ to find $x$ ${8.5x + 4.5}{(6)}{= 52.5}$ $8.5x+27 = 52.5$ $8.5x = 25.5$ $x = \dfrac{25.5}{8.5}$ ${x = 3}$ You can also plug ${y = 6}$ into $ {34x+9y = 156}$ and get the same answer for $x$ ${34x + 9}{(6)}{= 156}$ ${x = 3}$ There were $3$ teachers and $6$ students on the field trips.